Dickey Fuller Test- Random Walk
All random walks (with or without drift) have unit roots. The unit root problem occurs when absolute value of the lag coefficient equal to 1. A time series is stationary only, if absolute value of the lag coefficient is less than 1. If absolute value of the lag coefficient is greater than 1, then there is an explosive root.
If a time series appears to have a unit root, it is not covariance stationary. If a time series is not covariance stationary and we model it through AR process, the regression estimates will be spurious.
The Dickey and Fuller (1979) test is widely used to diagnose whether a time series has a unit root or not. The Dickey and Fuller is a regression-based test conducted on transformed versions of the AR(1) model, assuming the following models:
1)- Random Walk
2)- Random walk with drift
3)- Random walk with drift and trend
Random walk is a time series process where next period's value is obtained as the previous period's value, plus an unpredictable random error. We focus the first model here. The equation below is a special case of AR(1) model (excluding intercept term).
x t = b 1x t-1 + ∈ t.
Subtracting x t-1 from both sides gives.
x t - x t-1 = (b 1 - 1)x t-1 + ∈ t.
x t - x t-1 = \[ δ \]x t-1 + ∈ t.
Where, \[ δ \] = (b 1 - 1)
If b 1 = 1 then \[ δ \] = 0
Hence, a test of \[ δ \] = 0 is a test of b 1 = 1 .
We can formulate the following set of hypothesis.
H0 : \[ δ \] = 0 versus Ha : \[ δ \] ≠ 0
The null hypothesis of the test is that the time series contains a unit root and is nonstationary—and the alternative hypothesis is that the time series does not contain a unit root and is stationary.
In short, we regress the first-differenced time series (dependent variable) on the first lag of that time series (independent variable), where constant is zero.
The slope t-statistic obtained from the regression output is a statistic of the Dickey Fuller test, which is the basis for deciding whether or not to reject the null hypothesis, comparing it with the Dickey Fuller critical points rather than conventional t-critical values. The Dickey Fuller critical values are different for the above three models.
The comparison values we choose are based on the level of significance selected. The level of significance reflects how much sample evidence we require to reject the null. We can use three conventional significance levels to conduct hypothesis tests: 0.10, 0.05, and 0.01.
If we find that the calculated value of the test statistics is less than the Dickey Fuller t-critical value, we reject the null hypothesis.
Suppose, we decide to model the log of Starbucks Corporation's quarterly sales (in Million) using an AR (1) model.
Sales t = b 0 + b 1 ln Sales t-1 + ϵ t
Before, we estimate this regression, we should use the Dickey–Fuller test to determine whether there is a unit root in the log of SBUX’s quarterly sales. SBUX is the trade name of Starbucks Corporation on the New York Stock Exchange.
We use a short sample of quarterly data on SBUX’s sales from the first quarter of 2000 through the fourth quarter of 2007 (shown as default values in our calculator).
Run a Dickey Fuller Test at 0.05 Level of significance.
Dickey Fuller Test: Random Walk
We can use two different methods to correct the regression coefficients’ standard errors for serial correlation and such a correction may reverse the conclusions about a particular hypothesis test.
1- Robust standard errors (corrects the standard errors), also known as serial-correlation consistent standard errors or Hansen–White standard errors.
2- Generalized least squares (modifies the original equation), it requires econometric expertise to implement correctly on financial data.